Let $K$ be a convex body in $\mathbb{R}^n$, let $L$ be a lattice with covolume one, and let $\eta>0$. We say that $K$ and $L$ form an $\eta$-smooth cover if each point $x \in \mathbb{R}^n$ is covered by $(1 \pm \eta) vol(K)$ translates of $K$ by $L$. We prove that for any positive $\sigma, \eta$, asymptotically as $n \to \infty$, for any $K$ of volume $n^{3+\sigma}$, one can find a lattice $L$ for which $L, K$ form an $\eta$-smooth cover. Moreover, this property is satisfied with high probability for a lattice chosen randomly, according to the Haar-Siegel measure on the space of lattices. Similar results hold for random construction A lattices, albeit with a worse power law, provided the ratio between the covering and packing radii of $\mathbb{Z}^n$ with respect to $K$ is at most polynomial in $n$. Our proofs rely on a recent breakthrough by Dhar and Dvir on the discrete Kakeya problem.

翻译：设 $K$ 为 $\mathbb{R}^n$ 中的凸体，$L$ 为协体积为 1 的格，且 $\eta>0$。若每个点 $x \in \mathbb{R}^n$ 被 $K$ 的 $L$ 平移 $(1 \pm \eta) \mathrm{vol}(K)$ 次覆盖，则称 $K$ 与 $L$ 构成 $\eta$-光滑覆盖。我们证明：对任意正数 $\sigma, \eta$，当 $n \to \infty$ 时渐近地，对任意体积为 $n^{3+\sigma}$ 的 $K$，可找到格 $L$ 使得 $L, K$ 构成 $\eta$-光滑覆盖。此外，对按格空间上 Haar-Siegel 测度随机选取的格，该性质以高概率成立。关于随机构造 A 格亦得到类似结果（尽管幂次律更弱），前提是 $\mathbb{Z}^n$ 关于 $K$ 的覆盖半径与堆积半径之比至多为 $n$ 的多项式。我们的证明依赖于 Dhar 与 Dvir 在离散 Kakeya 问题上的最新突破。